Discrete Chaos, Second Edition: With

While maintaining the lucidity of the first edition, Discrete Chaos, Second Edition: With Applications in Science and Engineering now includes many recent results on global stability, bifurcation, chaos, and fractals. The first five chapters provide the most comprehensive material on discrete dynamical systems, including trace-determinant stability, bifurcation analysis, and the detailed analysis of the center manifold theory. This edition also covers L-systems and the periodic structure of the bulbs in the Mandelbrot set as well as new applications in biology, chemistry, and physics. The principal improvements to this book are the additions of PHASER software on an accompanying CD-ROM and the Maple'„¢ and MathematicaÂ® code available for download online.Incorporating numerous new topics and technology not found in similar texts, Discrete Chaos, Second Edition presents a thorough, up-to-date treatment of the theory and applications of discrete dynamical systems.

Preface	p. ix
Author Biography	p. xv
List of Symbols	p. xvii
Foreword	p. xix
The Stability of One-Dimensional Maps	p. 1
Introduction	p. 2
Maps vs. Difference Equations	p. 2
Maps vs. Differential Equations	p. 4
Euler's Method	p. 4
Poincare Map	p. 8
Linear Maps/Difference Equations	p. 9
Fixed (Equilibrium) Points	p. 14
Graphical Iteration and Stability	p. 19
Criteria for Stability	p. 25
Hyperbolic Fixed Points	p. 25
Nonhyperbolic Fixed Points	p. 28
Periodic Points and Their Stability	p. 36
The Period-Doubling Route to Chaos	p. 43
Fixed Points	p. 43
2-Periodic Cycles	p. 44
2[superscript 2]-Periodic Cycles	p. 46
Beyond [mu subscript infinity]	p. 50
Applications	p. 54
Fish Population Modeling	p. 54
Attraction and Bifurcation	p. 61
Introduction	p. 61
Basin of Attraction of Fixed Points	p. 62
Basin of Attraction of Periodic Orbits	p. 66
Singer's Theorem	p. 70
Bifurcation	p. 81
Sharkovsky's Theorem	p. 94
Li-Yorke Theorem	p. 94
A Converse of Sharkovsky's Theorem	p. 99
The Lorenz Map	p. 105
Period-Doubling in the Real World	p. 108
Poincare Section/Map	p. 110
	p. 110
Belousov-Zhabotinskii Chemical Reaction	p. 111
Appendix	p. 115
Chaos in One Dimension	p. 119
Introduction	p. 119
Density of the Set of Periodic Points	p. 120
Transitivity	p. 124
Sensitive Dependence	p. 129
Definition of Chaos	p. 137
Cantor Sets	p. 144
Symbolic Dynamics	p. 149
Conjugacy	p. 154
Other Notions of Chaos	p. 161
Rossler's Attractor	p. 163
Saturn's Rings	p. 167
Stability of Two-Dimensional Maps	p. 171
Linear Maps vs. Linear Systems	p. 171
Computing A[superscript n]	p. 172
Fundamental Set of Solutions	p. 179
Second-Order Difference Equations	p. 181
Phase Space Diagrams	p. 184
Stability Notions	p. 192
Stability of Linear Systems	p. 197
The Trace-Determinant Plane	p. 200
Stability Analysis	p. 200
Navigating the Trace-Determinant Plane	p. 204
Liapunov Functions for Nonlinear Maps	p. 207
Linear Systems Revisited	p. 215
Stability via Linearization	p. 219
The Hartman-Grobman Theorem	p. 224
The Stable Manifold Theorem	p. 225
Applications	p. 228
The Kicked Rotator and the Henon Map	p. 228
The Henon Map	p. 230
Discrete Epidemic Model for Gonorrhea	p. 233
Perennial Grass	p. 236
Appendix	p. 239
Bifurcation and Chaos in Two Dimensions	p. 241
Center Manifolds	p. 241
Bifurcation	p. 248
Eigenvalues of 1 or -1	p. 248
A Pair of Eigenvalues of Modulus 1 - The Neimark-Sacker Bifurcation	p. 249
Hyperbolic Anosov Toral Automorphism	p. 257
Symbolic Dynamics	p. 262
Subshifts of Finite Type	p. 263
The Horseshoe and Henon Maps	p. 268
The Henon Map	p. 272
A Case Study: The Extinction and Sustainability in Ancient Civilizations	p. 278
Appendix	p. 285
Fractals	p. 289
Examples of Fractals	p. 289
L-system	p. 299
The Dimension of a Fractal	p. 300
Iterated Function System	p. 314
Deterministic IFS	p. 315
The Random Iterated Function System and the Chaos Game	p. 325
Mathematical Foundation of Fractals	p. 330
The Collage Theorem and Image Compression	p. 338
The Julia and Mandelbrot Sets	p. 341
Introduction	p. 341
Mapping by Functions on the Complex Domain	p. 342
The Riemann Sphere	p. 351
The Julia Set	p. 354
Topological Properties of the Julia Set	p. 364
Newton's Method in the Complex Plane	p. 371
The Mandelbrot Set	p. 375
Topological Properties	p. 375
Rays and Bulbs	p. 377
Rotation Numbers and Farey Addition	p. 380
Accuracy of Pictures	p. 385
Bibliography	p. 389
Answers to Selected Problems	p. 397
Index	p. 414
Table of Contents provided by Ingram. All Rights Reserved.

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Discrete Chaos, Second Edition: With Applications in Science and Engineering

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